function res = fn_int_a1(vx, vf, k)

vp_tail = vx(end-9:end);
vf_tail = vf(end-9:end);
v2 = 1./vp_tail.^2;
v4 = 1./vp_tail.^4;
kc = vx(end);
[alpha, beta] = fn_fit(v2, v4, vf_tail);

vp0 = linspace(-4, 4, 800);
vp0 = vp0.^2.*sign(vp0);
vp1 = vp0 + k;
vp2 = vx;
vp3 = vp0 - k;

vp = [vp1, vp2, vp3];
vp(vp<0) = [];
vp(vp>kc) = [];
vp = unique(sort(vp));

vf2 = interp1(vx, vf, vp, 'pchip');
va = 1/2/pi./(1/4 + (k - vp).^2) + 1/2/pi./(1/4 + (k + vp).^2);

vnt = fn_nt(vp);
res = sum(vnt.*va.*vf2);

res = res + alpha*fn_ff2(k, kc) + beta*fn_ff4(k, kc);

end